Let $X$ be a smooth projective curve. Let $E$ be a stable vector bundle (i.e $\mu(F)< \mu(E)$ for all subbundles $F$). Let $L$ be any line bundle.
$G=E \otimes L$ is stable
I've computed the slope of $G$ to be $\mu(E)+degL$ . To proceed further I'd have to know what a subbundle of tensor product of vector bundles look like. How can I settle this?
[Here $\mu = deg/rank$]
We have $(E \otimes L) \otimes L^{-1} \cong E$, because tensor product of vector bundles is associative.
Given a subbundle $U \subset (E \otimes L)$ define $U' = U \otimes L^{-1} \subset (E \otimes L) \otimes L^{-1} \cong E$, a subbundle so by assumption $\mu(U \otimes L^{-1} ) < \mu(E)$ now using your tensor formula:
$\mu(U) = \mu(U \otimes L^{-1}) + \deg(L) < \mu(E) + \deg(L) = \mu(E \otimes L) .$
That is $E \otimes L$ is stable.